A three-parameter Jarratt-type iterative method for nonlinear systems with local convergence analysis, stability, and application

Purpose This study aims to develop a new three-parameter Jarratt-type iterative method for solving systems of nonlinear equations with higher order convergence, increased flexibility, and improved numerical performance compared to well-known existing iterative schemes. Design/methodology/approach The theoretical properties of the proposed iterative scheme are analysed using Taylor series expansions and a Lipschitz-type framework in Banach spaces, and an analytical expression for the radius of convergence is established. Stability and dynamic behaviour are investigated via basins of attraction in the cartesian plane and compared with some well-known existing methods. Findings The proposed method attains fourth-order convergence and exhibits superior efficiency and accuracy in numerical experiments. The radius of convergence is computed for several test problems. Comparative numerical results demonstrate faster convergence and enhanced robustness over existing methods, while basin-of-attraction analysis indicates a wider convergence region with fewer divergent points. Research limitations/implications The analysis is restricted to problems satisfying standard smoothness and Lipschitz continuity assumptions. Practical implications Due to its high convergence order, flexibility, and stability, the proposed iterative method is well suited for efficiently solving systems of nonlinear equations arising in applied mathematics, engineering, and scientific computing, including boundary value problems and nonlinear integral equations. Originality/value The originality of this work lies in the construction of a three-parameter Jarratt-type iterative method achieving fourth-order convergence, together with a rigorous convergence and radius analysis in Banach spaces. Furthermore, basin-of-attraction analysis offers new insights into the stability and robustness of the proposed scheme.